By AGT correspondence [1]}, Virasoro conformal blocks correspond to Nekrasov partition functions of 4d supersymmetric gauge theories. Hence, the tau functions of the Painlevé equations have explicit series expansions at regular singular points. Correspondence between partition functions of gauge theories and tau functions of Painlevé equations was investigated in [2]}. In [3]}, using Nakajima–Yoshioka blow-up equations [4]} for the pure Nekrasov partition functions, it was proved that the tau function of the Painlevé \(\mathrm {III}_3\)  (\(\mathrm {P_{III_3}}\) ) in terms of the pure Nekrasov partition function satisfies the bilinear equation for it.
